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b.
As analyzed in [
6
], the full metabolic network of the human red blood cell
involves 51 reactions (and their “fluxes,” in biochemical parlance) and 29metabo-
lites. What would be the shape of the stoichiometric matrix
S
for this metabolic
system
4
?
As one mathematical model,
5
one can also attempt to capture metabolic systems,
and biochemical reaction networks in general, pictorially, through graphs. In these
graphs, there is one vertex (i.e., node) for everymetabolite. Edges join twometabolites
if they are linked in a (bio)chemical reaction, and a value associated to each edge is
the “flux,” a measure of the level of activity through the reaction (or rate at which
the reaction is occurring). The direction of an arrow represents the direction of the
reaction, and reversible reactions are pictured through oppositely oriented double
edges (see Figure
8.4
). A large picture of multiple metabolic networks was given
earlier (Figure
8.1
); a simpler example of such a graph as given in [
6
] is pictured as
graph (A) in Figure
8.4
.
Such a graph as graph (A) in Figure
8.4
might correspond to looking at only part
of a larger system (just as glycolysis appears in the much larger network of cellular
metabolism). In this case, one can draw a boundary around the relevant part of the
system, and include arrows to represent flows into or out of the subsystem from the
larger system, pictured as graph
6
(B) of Figure
8.4
. There will be
internal fluxes
,
corresponding to reactions in the particular system under consideration, and
external
fluxes
, those involving inputs or outputs from parts of the system not under direct
focus, but that, given the connectedness of systems, cannot be ignored. In [
6
], it is
argued that a useful biochemical convention is to first represent the exchange fluxes
as arrows “going out” of the boundary, even if the direction in a chemical reaction
sense is the opposite. Hence, one sees the arrow conventions and labelings
7
of internal
fluxes (by
b
4
) in (B) of Figure
8.4
.
For each vertex in the graph of the biochemical reaction network, one may write
a “balance” or “node” equation in the internal and external fluxes. In this, the formal
sum of fluxes “going in” to a fixed node must equal the formal sum of fluxes “going
out.” Thus, using the second of the two graphs above (graph (B)) at node
A
, one
obtains
v
1
,...,v
7
) and external fluxes (
b
1
,...,
v
1
+
b
1
=
0
;
4
No computations are necessary to answer this, but see the final section for an associated project wherein
one can use free software to explore this example concretely, in more detail.
5
See the final section for the associated project which explores shortcomings and identifies another
model.
6
By many formal definitions, the result is no longer formally a graph in the mathematical sense, since
there are arrows which do not join two nodes, but we will continue to abuse terminology and call this
a graph.
7
This labeling of some fluxes by
v
i
s and some by
b
j
s is a notational convenience in what follows.
However, the total flux vector will still be denoted by
v
, with first seven entries the internal fluxes, and
last four entries the external fluxes.
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